PSI - Issue 16

Jaroslaw Galkiewicz / Procedia Structural Integrity 16 (2019) 35–42 Jaroslaw Galkiewicz / Structural Integrity Procedia 00 (2019) 000 – 000

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important factor is that this formula was developed for an inclusion in the form of a cylinder with infinite length, as shown in the study Neimitz and Janus (2016). Therefore, a reasonable solution is to observe the state of an inclusion at a known strain level and draw conclusions about critical stresses using the finite element method. Table 6 in Pineau and Pardoen (2007) is an example of gathering information about the level of critical stresses. This table reports only several values of critical stresses, at which a fracture or decohesion of the inclusion occurs. As the authors emphasize, this table has numerous deficiencies, such as the lack of information on the size of the inclusions and the density of their distributions, but it demonstrates well the difficulty of analysis of the void nucleation process. During analysis of the breaking process considering inclusion behavior, different strategies are used in practice. In the case of the Gurson-Needleman-Tvergard model, the nucleation process is described using a statistical approach to model the nucleation rate of new voids (Chu and Needleman (1990)): where f N is the volume fraction of nucleated voids, s N is the standard deviation of the nucleation of voids, ε N is the main strain of the nucleation of voids, ε p is the effective plastic strain and p  is the rate of the effective plastic strain. Unfortunately, determination of the parameters of the equation describing the nucleation of voids is highly problematic; therefore, the nucleation stage is usually omitted by assuming the initial void fraction in the material or modeling it directly in the analyzed model (Paroden and Hutchinson (2003), Hütter et al. (2014)). In many of the presented studies, the Gurson model was the basic analytical tool in which the basic idea consisted of modeling the average void fraction in the elementary cell. However, the cohesive model built on the fundamental works Barenblatt (1959), Leonov and Panasyuk (1959) is an equally good instrument for analysis of the state of stress, which accompanies breaking and debonding of inclusions. In this model, it is assumed that the surfaces of a material are bonded by cohesive forces of a known distribution. To separate the material, it is necessary to provide energy that is greater than the work of cohesive forces. The most important parameters in this model are the maximum stress in the cohesive zone and the cohesive work (Pineau and Pardoen (2007)). The shape of the cohesion curve depends on the fracture mechanism, which is modeled and is of secondary importance (Tvergaard and Hutchinson (1992)). The ABAQUS software assumes the simplest form of the cohesion curve, consisting of two straight branches describing the loading process and the weakening of the material after reaching the maximum stress level. The present study is not intended to analyze the behavior of a particular material, but the dimensions and mechanical properties of the inclusion and matrix materials are necessary for the construction of the elementary cell model. In the papers Faleskog et al. (1998), Gao et al. (1998), steel 21/4Cr1Mo was used for study. A detailed description of the structure of the material enabled construction of an elementary cell based on this steel. It contains large inclusions of manganese sulfide having sizes from 1 to 5 µm spaced at approximately 100 µm apart. This steel is characterized by a yield strength of 210 MPa and hardening exponent n=5. The Young’s modulus is 206 GPa. The critical value of the J-integral can be estimated based on J R curves at 350 kJ/m 2 . The described steel is a matrix material in the modeled elementary cell. The inclusion is composed of MnS. The inclusion material is treated as linear elastic such th at only the Young’s modulus and Poisson’s ratio are required for its characteristics. The study Juvonen (2004) reports that Young’s modulus in manganese sulfide changes in the range of 68 -138 GPa, while Poisson’s ratio is equal to 0.3. A value of the Young’s modulus equal to 100 GPa was chosen for the inclusion. 2. Material 2 exp 1 2   2 p      p N N nucl N N f f s s                 , (1)